Categories with strong monomorphic strong coimages
type: Item
instance of: scholarly article
title: Categories with strong monomorphic strong coimages
zbMATH Open document ID: 1251.18002
author: Juan-Miguel Gracia
published in: Missouri Journal of Mathematical Sciences
publication date: 2012-01-26T00:00:00Z
review text: Let \(\mathcal{C}\) be an arbitrary category and \(SE(\mathcal{C})\) (respectively, \(SM(\mathcal{C})\) be the subcategory of \(\mathcal{C}\) with the same objects and whose morphisms are strong epimorphisms (respectively, strong monomorphisms) of \(\mathcal{C}\). Also, let the pro-category pro-\(\mathcal{C}\) of \(\mathcal{C}\) be the universal category with directed inverse limits containing \(\mathcal{C}\) as a full subcategory. In this paper, the author investigates the question: when an object \(X\) of pro-\(\mathcal{C}\) is isomorphic to an object of pro-\(SE(\mathcal{C})\) or pro-\(SM(\mathcal{C})\). Moreover, he presents conditions which give stability of objects of pro-\(\mathcal{C}\). In this sense, the author considers \(\mathcal{C}\) a category with strong monomorphic strong coimages and obtains the following results: - if \(f:X\rightarrow Y\) is a strong epimorphism of pro-\(\mathcal{C}\) and X is isomorphic to an object of pro-\(SE(\mathcal{C})\), then \(Y\) is isomorphic to an object of pro-\(SE(\mathcal{C})\); - for an object \(X\) of pro-\(\mathcal{C}\), \(X\) is isomorphic to an object of pro-\(SM(\mathcal{C})\) if and only if there is a strong monomorphism \(f:X\rightarrow P\), where \(P\) is an object of \(\mathcal{C}\); - for \(P,Q\) objects of \(\mathcal{C}\), \(X\) object of pro-\(\mathcal{C}\), if \(f:P\rightarrow X\) is a strong epimorphism of pro-\(\mathcal{C}\) and \(g:X\rightarrow Q\) is a strong monomorphism of pro-\(\mathcal{C}\), then \(X\) is stable.
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 18A20
zbMATH DE Number: 6000843
zbMATH Keywords: category
zbMATH Keywords: (strong) monomorphism
zbMATH Keywords: (strong) epimorphism
zbMATH Keywords: strong coimages
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6000843